Percent Error Formula: Rules and Examples

We all make mistakes. Sometimes, if you play your cards right, they can become valuable learning opportunities. It's called "human error" for a reason; even the best of us leave a "t" uncrossed or an "i" undotted now and then. Such is life.

Before trying to correct a blooper, gaffe or snafu (did you know a thesaurus makes a great gift?), it's usually a good idea to find out what went wrong in the first place.

The size of the error is a key detail. How badly did you miss the mark? Was it a close shave or wildly off-base?

Picture a violinist in a philharmonic orchestra. On the night of a big concert, he misses an important cue and plays some notes too late. If he missed the cue by half a second, it might not be a huge deal. But if he missed it by half a minute, that's a different can of worms.

When there's a difference between the value you expected and the value you actually got — and you express that difference as a mathematical percentage — it's called a percent error or percentage error. Calculating percentage error involves comparing an expected value and an actual value to determine how far reality deviated from theoretical expectations.

Today, we're going to take the mystery out of reporting the percent error correctly and show you how to use it in real life.

What Is the Percent Error Formula?

The equation couldn't be much simpler. Here it is:

Percent Error = | Experimental Value – Actual Value | / Actual Value x 100%

The value you originally projected goes by many names, including exact value, accepted value, estimated value, theoretical value, approximate value or experimental value, depending on the context. For instance, a physics student calculating speed will refer to an accepted value based on the velocity formula, but the approximate or measured value of speed in her experiment may differ. In another scenario, a business owner may refer to an estimated value when forecasting revenue.

Similarly, there are multiple labels for your real-life result, including actual value, measured value, and exact or known value. No matter what you call it, the spirit behind the actual number itself remains the same.

Some people find written instructions more helpful than mathematical formulas. If you're one of them, no worries. Here's a step-by-step walkthrough for percent error calculation:

Step One: Take the experimental value and subtract the actual value from it. This is called the relative error.
Step Two: Take the absolute value of the number you arrived at in Step One (that's what those two vertical lines indicate). This new number is called the absolute error and ensures that your final percentage does not include a negative sign.
Step Three: Divide that number by the actual value.
Step Four: Multiply your result by 100.
Step Five: Write out your final answer as a percentage.

Examples of How to Calculate Percent Error

We're now ready to take the percentage error formula out for a test drive.

Example 1

Let's say you're a bookworm with a long vacation coming up. You go to the library to grab some reading material. Before you open the front door, you assume you'll check out three books. But instead, for whatever reason, you only take home two books. What's the percentage error of your estimate?

In our example, the experimental value is 3 and the actual value is 2. Plug in the numbers, and you get this:

Percent Error = (3 – 2)/2 x 100

If you're old enough to read this article, we're guessing you already knew that 3 minus 2 equals 1. Which leaves us with:

Percent Error = 1/2 x 100

Divide 1 by 2 and you get the following:

Percent Error = 0.5 x 100

And 100 times 0.5 equals 50. But remember, we have to express our final answer as a percentage. When we do that, we learn the original guess you made had a percent error of 50%.

This example was all about quantity (i.e., the number of library books). But the percent error formula can also be applied to lots of other values — like speed, distance, mass and time.

Bearing that in mind, let's go through the formula again.

Example 2

Suppose a college athlete thinks he'll need 45 seconds to finish a hardcore workout challenge. But when he hits the gym, the routine takes him 60 seconds to complete. What was the percent error of the time estimate he started out with (45 seconds)?

Percent error = (45 – 60)/60 x 100

Right off the bat, we've hit a complication. If you subtract 60 from 45, you get a negative number (-15 to be exact).

Divide -15 by 60 and you'll get -0.25, which is another negative value. And we can't stop there; we still need to multiply the -0.25 by 100, giving us an answer of -25. Does that mean the percent error is -25%?

The percent error between an estimated value and the actual value cannot be expressed as a negative. It's always written out as a positive value, whether the starting estimate was way too big or way too small.

Here's where our old friends "absolute error" and "relative error" come into play. The value of -15 is only the relative error. You need to take the absolute value of that before proceeding with the calculation. Once you have the absolute error of 15, you can divide that by 60 and multiply by 100 for a percent error of 25%.